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GEOMETRICAL RESEARCHES 


THE THEORY OF PARALLELS. 


BY 


NICHOLAUS LOBATSCHEWSKY, 


IMPERIAL RUSSIAN REAL COUNCILLOR OF STATE AND REGULAR PROFESSOR 


OF MATHEMATICS IN THE UNIVERSITY OF KASAN. 


BERLIN, 1840. 


TRANSLATED FROM THE ORIGINAL 
BY 
GEORGE BRUCE HALSTED, 


A. M., Ph. D., Ex-Fellow of Princeton College and Johns Hopkins University, 
Professor of Mathematics in the University of Texas. 


AUSTIN: 
PUBLISHED BY THE UNIVERSITY OF TEXAS, 
1891. 





TRANSLATORS PREFACE. 





Lobatschewsky was the first man ever to publish a non-Euclidian geom- 
etry. 

Of the immortal essay now first appearing in English Gauss said, ‘‘The 
author has treated the matter with a master-hand and in the true geom- 
eter’s spirit. I think I ought to call your attention to this book, whose 
perusal can not fail to give you the most vivid pleasure.” 

Clifford says, ‘It is quite simple, merely Euclid. without the vicious 
assumption, but the way things come out of one another is quite lovely.” 
* * * «What Vesalius was to Galen, what Copernicus was to Ptolemy, 
that was Lobatschewsky to Euclid.” 

Says Sylvester, “In Quaternions the example has been given of Al- 
gebra released from the yoke of the commutative principle of multipli- 
cation—an emancipation somewhat akin to Lobatschewsky’s of Geometry 
from Euclid’s noted empirical axiom.” 

Cayley says, “It is well known that Huclid’s twelfth axiom, even in 
Playfair’s form of it, has been considered as needing demonstration; 
and that Lobatchewsky constructed a perfectly consistent theory, where- 
in this axiom was assumed not to hold good, or say a system of non- 
Euclidian plane geometry. There is a like system of non-Euclidian solid 
geometry.” 

GEORGE BRUCE HALSTED. 

2407 San Marcos Street, 

Austin, Texas. 

May 1, 1891. 


[3] 





TRANSLATORS INTRODUCTION. 





“Prove all things, hold fast that which is good,” does not mean dem- 
onstrate everything. rom nothing assumed, nothing can be proved. 
“Geometry without axioms,” was a book which went through several 
editions, and still has historical value. But now a volume with such a 
title would, vithout opening it, be set down as simply the work of a 
paradoxer. 

The set of axioms far the most influential in the intellectual history 
of the world was put together in Egypt: but really it owed nothing. to 
the Egyptian race, drew nothing from the boasted lore of Hgypt’s 
priests. 

The Papyrus of the Rhind, belonging to the British Museum, but 
given to the world by the erudition of a German Egyptologist, Hisen- 
lohr, and a German historian of mathematics, Cantor, gives us more 
knowledge of the state of mathematics in ancient Egypt than all else 
previously accessible to the modern world. Its whole testimony con- 
firms with overwhelming force the position that Geometry as a science, 
strict and self-conscious deductive reasoning, was created by the subtle 
intellect of the same race whose bloom in art still overawes us in the 
Venus of Milo, the Apollo Belvidere, the Laocoon. 

In a geometry occur the most noted set of axioms, the geometry of 
Huclid, a pure Greek, professor at the University of Alexandria. 

Not only at its very birth did this typical product of the Greek genius 
assume sway as ruler in the pure sciences, not only does its first efflor- 
escence carry us through the splendid days of Theon and Hypatia, but 
unlike the latter, fanatics can not murder it; that dismal flood, the dark 
ages, can not drown it. Like the pheenix of its native Egypt, it rises 
with the new birth of culture. An Anglo-Saxon, Adelard of Bath, 
finds it clothed in Arabic vestments in the land of the Alhambra. Then 
clothed in Latin, it and the new-born printing press confer honor on 
each other. Tinally back again in its original Greek, it is published 
first in queenly Venice, then in stately Oxford, since then everywhere. 
The latest edition in Greek is just issuing from Leipsic’s learned presses. 

[5] 


6,\- THEORY OF PARALLELS. 


How the first translation into our cut-and-thrust, survival-of-the-fittest 
English was made from the Greek and Latin by Henricus Billingsly, 
Lord Mayor of London, and published with a preface by John Dee the 
Magician, may be studied in the Library of our own Princeton College, 
where they have, by some strange chance, Billingsly’s own copy of the 
Latin version of Commandine bound with the Editio Princeps in Greek 
and enriched with his autograph emendations. [ven to-day in the vast 
system of examinations set by Cambridge, Oxford, and the British gov- 
ernment, no proof will be accepted which infringes Euclid’s order, a 
sequence founded upon his set of axioms. ; 

The American ideal is success. In twenty years the American maker 
expects to be improved upon, superseded. The Greek ideal was per- 
fection. The Greek Hpic and Lyric poets, the Greek sculptors, remain 
unmatched. The axioms of the Greek geometer remained unquestioned 
for twenty centuries. 

How and where doubt came to look toward them is of no ordinary 
interest, for this doubt was epoch-making in the history of mind. 

Among Euclid’s axioms was one differing from the others in pro- 
lixity, whose place fluctuates in the manuscripts, and which is not used 
in Kuclid’s first twenty-seven propositions. Moreover it is only then 
brought in to prove the inverse of one of these already demonstrated. 

All this suggested, at Hurope’s renaissance, not a doubt of the axiom, 
but the possibility of getting along without it, of deducing it from the 
other axioms and the twenty-seven propositions already proved. Huclid 
demonstrates things more axiomatic by far. He proves what every dog 
knows, that any two sides of a triangle are together greater than the 
third. Yet when he has perfectly proved that lines making with a 
transversal equal alternate angles are parallel, in order to prove the in- 
verse, that parallels cut by a transversal make equal alternate angles, he 
brings in the unwieldly postulate or axiom: 

“Tf a straight line meet two straight lines, so as to make the two in- 
terior angles on the same side of it taken together less, than two right 
angles, these straight lines, being continually produced, shall at length 
meet on that side on which are the angles which are less than two right 
angles.” 

Do you wonder that succeeding geometers wished by demonstration 
to push this unwieldly thing from the set of fundamental axioms. 


TRANSLATOR’S INTRODUCTION. if 


Numerous and desperate were the attempts to deduce it from reason- 
ings about the nature of the straight line and plane angle. In the 
‘“‘Hincyclopceedie der Wissenschaften und Kunste; Von Ersch und Gru- 
ber;” Leipzig, 1838; under ‘Parallel,’ Sohncke says that in mathe- 
matics there is nothing over which so much has been spoken, written, 
and striven, as over the theory of parallels, and all, so far (up to his 
time), without reaching a definite result and decision. 

Some acknowledged defeat by taking a new definition of parallels, as 
for example the stupid one, ‘“ Parallel lines are everywhere equally dis- 
tant,” still given on page 33 of Schuyler’s Geometry, which that author, 
like many of his unfortunate prototypes, then attempts to identify with 
Huclid’s definition by pseudo-reasoning which tacitly assumes Euclid’s 
postulate, e. g. he says p. 35: ‘Tor, if not parallel, they are not every- 
where equally distant; and since they lie in the same plane; must ap- 
proach when produced one way or the other; and since straight lines 
continue in the same direction, must continue to approach if produced 
farther, and if sufficiently produced, must meet.” This is nothing but 
HKuclid’s assumption, diseased and contaminated by the introduction of 
the indefinite term ‘ direction.” 

How much better to have followed the third Class of his predecessors 
who honestly assume a new axiom differing from Huclid’s in form if 
not in essence. Of these the best is that called Playfair’s; “Two lines 
which intersect can not both be parallel to the same line.” 

The German article mentioned is followed by a carefully prepared 
list of ninety-two authors on the subject. In English an account of 
like attempts was given by Perronet Thompson, Cambridge, 1833, and 
is brought up to date in the charming volume, “ Huclid and his Modern 
Rivals,” by C. L. Dodgson, late Mathematical Lecturer of Christ Church, 
Oxford. 

All this shows how ready the world was for the extraordinary flaming- 
forth of genius from different parts of the world which was at once to 
overturn, explain, and remake not only all this subject but as conse- 
quence all philosophy, all ken-lore. As was the case with the dis- 
covery of the Conservation of Energy, the independent irruptions 
of genius, whether in Russia, Hungary, Germany, or even in Canada 
gave everywhere the same results. 

At first these results were not fully understood even by the brightest 


8 THEORY OF PARALLELS. 


intellects. Thirty years after the publication of the book he mentions, 
we see the brilliant Clifford writing from Trinity College, Cambridge, 
April 2, 1870, ““Several new ideas have come to me lately: First I 
have procured Lobatschewsky, ‘Etudes Geometriques sur la Theorie 
des Parallels’ -— -— -— asmall tract of which Gauss, therein quoted, 
says: L’auteura traite la matiere en main de maitre et avec le veritable 
esprit geometrique. Je crois devoir appeler votre attention sur ce livre, 
dont la lecture ne peut manquer de vous causer le plus vif plaisir.’” 
Then says Clifford: ‘It is quite simple, merely Euclid without the 
vicious assumption, but the way the things come out of one another is 
quite lovely.” 3 

The first axiom doubted is called a ‘vicious assumption,” soon no 
man sees more clearly than Clifford that all are assumptions and none 
vicious. He had been reading the French translation by Houel, pub- 
lished in 1866, of a little book of 61 pages published in 1840 in Berlin 
under the title Geometrische Untersuchungen zur Theorie der Parallel- 
linien by a Russian, Nicolaus Ivanovitch Lobatschewsky (1793-1856), 
the first public expression of whose discoveries, however, dates back to 
a discourse at Kasan on February 12, 1826. 

Under this commonplace title who would have suspected the dis- 
covery of a new space in which to hold our universe and ourselves. 

A new kind of universal space; the idea is a hard one. To name it, 
all the space in which we think the world and stars live and move and 
have their being was ceded to Euclid as his by right of pre-emption, 
description, and occupancy; then the new space and its 3 quieks -following 
fellows could be called Non-Huclidean. 

Gauss in a letter to Schumacher, dated Nov. 28, 1846, mentions that 
as far back as 1792 he had started on this path to a new universe. 
Again he says: ‘La Geometrie non-EHuclidienne ne renferme en elle 
rien de contradictoire, quoique, a premiere vue, beaucoup de ses resul- 
tats aien l’air de paradoxes. Ces contradictions apparents doivent etre 
regardees comme l’effet d’une illusion, due a l’habitude que nous avons 
prise de pue heure de considerer la geometrie Huclidienne comme 
rigoureuse. ” 

But here we see in the last word the same imperfection of view as in 
Clifford’s letter. The perception has not yet come that though the non- 
Euclidean geometry is rigorous, Euclid is not one whit less so. 


TRANSLATORS INTRODUCTION, 9 


A clearer idea here had already come to the former room-mate of 
_ Gauss at Goettingen, the Hungarian Wolfgang Bolyai. His principal 
work, published by subscription, has the following title: 

Tentamen Juventutem studiosam in elementa Matheseos purae, ele- 
mentaris ac sublimioris, methodo intuitiva, evidentique -huic propria, in- 
troducendi. Tomus Primus, 1831; Secundus, 1833. 8vo. Maros-Va- 
sarhelyini. 

In the first volume with special numbering, appeared the celebrated 
Appendix of his son Johann Bolyai with the following title: 

Ap., scientiam spatii absolute veram exhibens: a veritate aut falsitate 
Axiomatis XI Euclidei (a priori haud unquam decidenda) independen- 
tem. Auctore Johanne Bolyai de eadem, Geometrarum in Exercitu 
Caesareo Regio Austriaco Castrensium Captaneo. Maros-Vasarhely., 
1832. (26 pages of text). 

This marvellous Appendix has been translated into French, Italian, 
and German. 

In the title of Wolfgang Bolyai’s last work, the only one he com- 
posed in German (88 pages of text, 1851), occurs the following: 

“Und da die Frage, ob zwei von der dritten geschnittene Geraden wenn die 
Summa der inneren Winkel nicht = 2R, sich schnerden oder nicht?, niemand 
auf der Erde ohne ein Axiom (wie Euclid das XI) aufzustellen, beant- 
worten wird; die davon unabhengige Geometrie abzusondern, und 
eine auf die Ja Antwort, andere auf das Nein so zu bauen, dass die 
Formeln der letzen auf ein Wink auch in der ersten gultig seien.” 

The author mentions Lobatschewsky’s Geometrische Untersuchungen 
Berlin, 1840, and compares it with the work of his son Johann Bolyai, 
‘‘an sujet duquel il dit- ‘Quelques exemplaires de l’ouvrage publie ici 
ont ete envoyes a cette epoque a Vienne, a Berlin, a Gettingen. . . De 
Goettingen le geant mathematique, [Gauss] qui du sommet des hauteurs 
embrasse du meme regard les astres et la profondeur des abimes, a ecrit 
qu'il etait ravi de voir execute le travail qu’il avait commence pour le 
laisser apres lui dans ses papiers.’”’ 

Yet that which Bolyai and Gauss, a mathematician never surpassed 
in power, see that no man can ever do, our American Schuyler, in the 
density of his ignorance, thinks that he has easily done. 

In fact this first of the Non-Euclidean geometries accepts all of Eu- 
clid’s axioms but the last, which it flatly denies and replaces by its con- 


aa 


10 - THEORY OF PARALLELS. 


tradictory, that the sum of the angles made on the same side of a trans- 
versal by two straight lines may be less than a straight angle without 
the lines meeting. <A perfectly consistent and elegant geometry then 
follows, in which the sum of the angles of a triangle is always less than 
a straight angle, and not every triangle has its vertices concyclic. 


THEORY OF PARALLELS. 


In geometry I find certain imperfections which I hold to be the rea- 
son why this science, apart from transition into analytics, can as yet 
make no advance from that state in which it has come to us from Euclid. 

As belonging to these imperfections, I consider the obscurity in the 
fundamental concepts of the geometrical magnitudes and in the manner 
and method of representing the measuring of these magnitudes, and 
finally the momentous gap in the theory of parallels, to fill which all ef- 
forts of mathematicians have been so far in vain. 

For this theory Legendre’s endeavors have done nothing, since he 
was forced to leave the only rigid way to turn into aside path and take 
refuge in auxiliary theorems which he illogically strove to exhibit as 
necessary axioms. My first essay on the foundations of geometry I pub- 
lished in the Kasan Messenger for the year 1829. In the hope of having 
satisfied all requirements, I undertook hereupon a treatment of the whole 
of this science, and published my work in separate parts in the ‘“Ge- 
lehrten Schriften der Universitet Kasan” for the years 1836, 1837, 1838, 
under the title “New Elements of Geometry, with a complete Theory 
of Parallels.” The extent of this work perhaps hindered my country- 
men from following such a subject, which since Legendre had lost its 
interest. Yet [ am of the opinion that the Theory of Parallels should 
not lose its claim to the attention of geometers, and therefore I aim to 
give here the substance of my investigations, remarking beforehand that 
contrary to the opinion of Legendre, all other imperfections— for ex- 
ample, the definition of a straight line—show themselves foreign here 
and without any real influence on the theory of parallels. 

In order not to fatigue my reader with the multitude of those theo- 
rems whose proofs present no difficulties, I prefix here only those of 
which a knowledge is necessary for what follows. 

1. A straight line fits upon itself in all its positions. By this I mean 
- that during the revolution of the surface containing it the straight line 
does not change its place if it goes through two unmoving points in the 
surface: (7. e., if we turn the surface containing it about two points of 


the line, the line does not move.) 
. [11] 


12 THEORY OF PARALLELS. 


2. Two straight lines can not intersect in two points. 

3. <A straight line sufficiently produced both ways must go out 
beyond all bounds, and in such way cuts a bounded plain into two parts. 

4, Two straight lines perpendicular to a third never intersect, how 
far soever they be produced. 3 mia 

5. A straight line always cuts another in going from one side of it 
over to the other side: (¢. e, one straight line must cut another if it 
has points on both sides of it.) 

6. Vertical angles, where the sides of one are productions of the 
sides of the other, are equal. This holds of plane rectilineal angles, 
among themselves, as also of plane surface angles: (7. e., dihedral angles.) 

7. Two straight lines can not intersect, if a third cuts them at the 
same angle. 


8. Ina rectilineal triangle equal sides lie opposite equal angles, and 
inversely. 


9. In a rectilineal triangle, a greater side lies opposite a greater 
angle. In aright-angled triangle the hypothenuse is greater than either 
of the other sides, and the two angles adjacent to it are acute. 

10. Rectilineal triangles are congruent if they have a side and two 
‘angles equal, or two sides and the included angle equal, or two sides and 
the angle opposite the greater equal, or three sides equal. 

11. A straight line which stands at right angles upon two other 
straight lines not in one plane with it is perpendicular to all straight 
lines drawn through the common intersection point in the plane of those 
two. 

12. The intersection of a sphere with a’plane is a circle. 

18. <A straight line at right angles to the intersection of two per- 
pendicular planes, and in one, is perpendicular to the other. 

14. In aspherical triangle equal sides lie opposite equal angles, and 
inversely. 

15. Spherical triangles are congruent (or symmetrical) if they have 
two sides and the included angle equal, or a side and the adjacent angles 
equal. 

From here follow the other theorems with their explanations and 


* 


proofs. 


THEORY OF PARALLELS. 13 


16. All straight lines which in a plane go out from a point can, 
with reference to a given straight line in the same plane, be divided 
into two classes —into cutting and not-cutting. 

The boundary lines of the one and the other class of those lines will 
be called parallel to the given line. 

From the point A (Fig. 1) let fall upon the 
line BC the perpendicular AD, to which again 
draw the perpendicular AE. . 

In the right angle HAD either will all straight 
lines which go out from the point A meet the 
line DC, as for example AF, or some of them, 
like the perpendicular AE, will not meet the 
line DC. In the uncertainty whether the per- 
pendicular AH is the only line which does not 





meet DO, we will assume it may be possible that 
there are still other lines, for example AG, Via. 1. 

which do not cut DC, how far soever they may be prolonged. In pass- 
ing over from the cutting lines, as AF’, to the not-cutting lines, as AG, 
we must come upon a line AH, parallel to DO, a boundary line, upon 
one side of which all lines AG are such as do not meet the line DC, 
while upon the other side every straight line AF cuts the line DC. 

The angle HAD between the parallel HA and the perpendicular AD 
is called the parallel angle (angle of parallelism), which we will here 
designate by // (p) for AD = p. 

If // (p) is a right angle, so will the prolongation AH’ of the perpen- 
dicular AE likewise be parallel to the prolongation DB of the line DC, 
in addition to which we remark that in regard to the four right angles, 
which are made at the point A by the perpendiculars AE and AD, 
and their prolongations AK’ and AD’, every straight line which goes 
out from the point A, either itself or at least its prolongation, lies in one 
of the two right angles which are turned toward BC, so that except the 
parallel HE’ all others, if they are sufficiently produced both ways, must 
intersect the line BC. | 

If JT (p) < 47, then upon the other side of AD, making the same 
angle DAK = // (p) will he also a line AK, parallel to the prolonga- 
tion DB of the line DC, so that under this assumption we must also - 
make a distinction of sides in parallelism. 


14 THEORY OF PARALLELS. 


All remaining lines or their prolongations within the two right angles 
turned toward BC pertain to those that intersect, if they lie within the 
angle HAK = 2 J] (p) between the parallels; they pertain on the other 
_hand to the non-intersecting AG, if they lie upon the other sides of the 


parallels AH and AK, in the opening of the two angles EAH = 47. 


— I1(p), E’AK=47- — JI (p), between the parallels and EH’ the per- 
pendicular to AD. Upon the other side of the perpendicular EE’ will 
in like manner the prolongations AH’ and AK’ of the parallels AH and 
AK likewise be parallel to BC; the remaining lines pertain, if in the 
angle K’AH’, to the intersecting, but if in the angles K’AH, H’AH’ 
to the non-intersecting. 

In accordance with this, for the assumption //(p) = 4. the lines can 
be only intersecting or parallel; but if we assume that /7(p) < $7, then 
we must allow two parallels, one on the one and one on the other side; 
in addition we must distinguish the remaining lines into non-intersect- 
ing and intersecting. 

For both assumptions it serves as the mark of parallelism that the 
line becomes intersecting for the smallest deviation toward the side 
where lies the parallel, so that if AH is parallel to DC, every line AF 
cuts DC, how small soever the angle HAT’ may be. 





THEORY OF PARALLELS. 15 


17. A straight line maintains the characteristic of parallelism at all its 
pownts. 
Given AB (Fig. 2) parallel to CD, to which latter AC is perpendic 





ular. We will consider two points taken at random on the line AB and 
its production beyond the perpendicular. 

Let the point E lie on that side of the perpendicular on which AB is 
looked upon as parallel to CD. 

Let fall from the point E a perpendicular EK on CD and so draw EF 
that it falls within the angle BEK. 

Connect the points A and F by a straight line, whose production then 
(by Theorem 16) must cut CD somewhere in G. Thus we get a triangle 
ACG, into which the line EF goes; now since this latter, from the con. 
struction, can not cut AC, and can not cut AG or EK a second time 
(Theorem 2), therefore it must meet CD somewhere at H (Theorem 3). 

Now let H’ be a point on the production of AB and E’K’ perpendic- 
ular to the production of the line CD; draw the line E’F’ making so 
small an angle AE’H” that it cuts AC somewhere in F’; making the 
same angle with AB, draw also from A the line AI’, whose production 
will cut CD in G (Theorem 16.) 

Thus we get a triangle AGC, into which goes the production of the 
line E’F’; since now this line can not cut AH a second time, and also 
can not cut AG, since the angle BAG = BE’G’, (Theorem 7), therefore 

must it meet CD somewhere in G’. 

Therefore from whatever points EK and H’ the lines HF and H’F”’ go 
out, and however little they may diverge from the line AB, yet will 
they always cut CD, to which AB is parallel. 


16 THEORY OF PARALLELS. 


18. Two lines are always mutually parallel, _ 
Let AC be a perpendicular on CD, to which AB is parallel 
if we draw from © the line R 





CE making any acute angle 
ECD with CD, and let fall 
from A the perpendicular AF 
upon CE, we obtain a right- 
angled triangle ACF, in which 
AC, being the hypothenuse, 

is greater than the side AF + 
(Theorem 9.) Cc K 

Make AG = AF’, and slide Bie. 3. 
the figure EFAB until AF coincides with AG, when AB and FE aa 
take the position AK and GH, such that the angle BAK’= FAC, con- 
sequently AK must cut the line DC somewhere in K (Theorem 16), thus 
forming a triangle AKC, on one side of which the perpendicular GH 
intersects the line AK in L (Theorem 3), and thus determines the dis- 
tance AL of the intersection point of the lines AB and CE on the line 
AB from the point A. 

Hence it follows that CE will always intersect AB, how small soever 
may be the angle ECD, consequently CD is parallel to AB (Theorem 16.) 

19. In a rectilineal triangle the sum of the three angles can not be greater 
than two right angles. 

Suppose in the triangle ABC (Fig. 4) the sum of the three angles is 
equal to z + a; then choose in case : 
of the inequality of the sides the 
smallest BO, halve it in D, draw 2 
from A through D the line AD 
and make the prolongation of. it, 

DE, equal to AD, then join the 4 c 

point E to the point C by the BIG. 2 

straight line HC. In the congruent triangles ADB and CDH, ibe angle 
ABD = DCE, and BAD = DEC (Theorems 6 and 10); whence follows 
that also in the triangle ACE the sum of the three angles must be equal 
to z-+a; but also the smallest angle BAC (Theorem 9) of the triangle 
ABC in passing over into the new triangle ACH has been cut up into 
the two parts HAC and AEC. Continuing this process, continually 


| 
| 
\ 
| 
| 


| 


THEORY OF PARALLELS, 1% 


halving the side opposite the smallest angle, we must finally attain to a 
triangle in which the sum of the three angles is z + a, but wherein are 
two angles, each of which in absolute magnitude is less than 4a; since 
now, however, the third angle can not be greater than z, so must a be 
either null or negative, 

20. If in any rectilineal triangle the sum of the three angles is equal to 
two right angles, so is this also the case for every other triangle. 

If in the rectilineal triangle ABC (Fig. 5) the sum of the three angles 
= xz, then must at Icast two of its angles, A B 
and OC, be acute. Let fall from the vertex of 
the third angle B upon the opposite side AC 
the perpendicular p. This will cut the tri- A 
angle into two right-angled triangles, in each Fie. 5. 
of which the sum of three angles must also be 7z, since it can not in 
either be greater than z, and in their combination not less than z. . 

So we obtain a right-angled triangle with the perpendicular sides p 
and q, and from this a quadrilateral whose opposite sides are equal and 
whose adjacent sides p and q are at right angles (Fig. 6.) 

By repetition of this quadrilateral we can make another with sides 
np and q, and finally a quadrilateral ABCD with sides at right angles 
to each other, such that AB = np, AD = mq, DC = np, BC = mq, where 





m and n are any whole numbers. Such a quadrilateral is divided by 
the diagonal DB into two congruent right-angled triangles, BAD and 
BCD, in each of which the sum of the three angles = z. 

The numbers n and m can be taken sufficiently great for the right- 
angled triangle ABC (Fig. 7) whose perpendicular sides AB = np, BC 
= mq, to enclose within itself another given (right-angled) triangle BDE 


as soon as the right-angles fit each other. 
2— par. 


18 THEORY OF PARALLELS. 


Drawing the line DC, we obtain right-angled triangles of which every 
successive two have a side in common. 

The triangle ABC is formed by the union of the two triangles ACD 
and DCB, in neither of which can the sum of the angles be greater than 
z; consequently it must be equal to z, in order that the sum in the 
compound triangle may be equal to z. 


A 4 B 
Fie. 7. 

In the same way the triangle BDC consists of the two triangles DEC 
and DBE, consequently must in DBE the sum of the three angles be 
equal to z, and in general this must be true for every triangle, since 
each can be cut into two right-angled triangles. 

From this it follows that only two hypotheses are allowable: Either 
is the sum of the three angles in all rectilineal triangles equal to z, or 
this sum is in all less than z. . 

21. From a given point we can always draw a straight line that shall 
make with a given straight line an angle as small as we choose. 

Let fall from the given point A (Fig. 8) upon the given line BC the 

4 


Wa. 8. 
perpendicular AB; take upon BC at random the point D; draw the line 
AD; make DE = AD, and draw AH. 


THEORY OF PARALLELS. 19 


In the right-angled triangle ABD let the angle ADB = a; then must 
in the isosceles triangle ADE the angle AED be either 4a or less (Theo- 
rems 8 and 20). Continuing thus we finally attain to such an angle, 
AXB, as is less than any given angle. 

22. If two perpendiculars to the same straight line are parallel to each 
other, then the sum of the three angles in a rectilineal triangle is equal to two 
right angles. 

Let the lines AB and CD (Fig. a be parallel to each other and per- 
pendicular to AC. B 

Draw from A the lines AH 
and AF to the points E and F, 
which are taken on the line CD 
at any distances F'C > EC from 
the point C. c 

Suppose in the right-angled tri- 
angle ACE the sum of the three angles is equal to z — 4, in the tri- 
angle AHF equal to z — §, then must it in triangle ACF equal z — « 
— f, where z and f can not be negative. 

Further, let theangle BAF = a, AFC = b, sois g++ 8 =a — b; now 
by revolving the line AF away from the perpendicular AC we can make 
the angle a between AF and the parallel AB as small as we choose; so 











E ¥ 
Fia. 9. 


also can we lessen the angle b, consequently the two angles x and 2 
can have no other magnitude than 4 = 0 and § = 0. 

It follows that in all rectilineal triangles the sum of the three angles 
is either z and at the same time also the parallel angle /7 (p) = 4 x for 
every line p, or for all triangles this sum is < z and at the same time 
also /[(p)< tz 

The first ear serves as foundation for the ordinary geometry and 
plane trigonometry. 

The second assumption can likewise be admitted without leading to 
any contradiction in the results, and founds a new geometric science, 
to which I have given the name Jmaginary Geometry, and which I in- 
tend here to expound as far as the development of the equations be- 
tween the sides and angles of the rectilineal and spherical triangle. 

23. Sor every given angle ¢ we can find a line p such that IT (p) = «. 

| Let AB and AC (fig. 10) be two straight lines which at the inter- 
| section point A make the acute angle z; take at random on AB a point 


20 THEORY OF PARALLELS. 


B’; from this point drop B’A’ at right angles to AC; make A’A” — 
AA’; erect at A” the perpendicular A”B’; and so continue until a per- 





N A” K F C 


Fig. 10. 


pendicular CD is attained, which no longer intersects AB. This must 
of necessity happen, for if in the triangle AA’B’ the sum of all three 


angles is equal to z — a, then in the triangle AB’ A” it equals z — 2a, 
in triangle AA’B” less than z — 2a (Theorem 20), and so forth, until 


it finally becomes negative and thereby shows the impossibility of con- 
structing the triangle. 

The perpendicular CD may be the very one nearer than which to the 
point A all others cut AB; at least in the passing over from those that 
cut to those not cutting such a perpendicular FG must exist. 

Draw now from the point F the line FH, which makes with FG the 
acute angle HIG, on that side where lies the point A. From any point 
H of the line FH let fall upon AC the perpendicular HK, whose pro- 
longation consequently must cut AB somewhere in B, and so makes a 
triangle AKB, into which the prolongation of the line FH enters, and 
therefore must meet the hypothenuse AB somewhere in M. Since the 
angle GFH is arbitrary and can be taken as small as we wish, therefore 
¥G is parallelto AB and AF =p. (Theorems 16 and 18.) 

One easily sees that with the lessening of pthe angle z increases, while, 
for p == 0, it approaches the value 47; with the growth of p the angle 
a decreases, while it continually approaches zero for p =o. 

Since we are wholly:at liberty to choose what angle we will under- 


THEORY OF PARALLELS. yA 


stand by the symbol // (p) when the line p is expressed by a negative 
number, so we will assume 
I(p)+- 1 —p)=-, 
an equation which shall hold for all values of p, positive as well as neg- 
ative, and for p= 0. : 
24. The farther parallel lines are prolonged on the side of their paral- 
lelism, the more they approach one another. 


If to the line- AB (Fig. 11) two perpendiculars AC = BE are erected 


and their end-points Cand HE joined by « F F 
a straight line, then will the quadrilat- 
eral CABE have two right angles at Ms 
A and B, but two aeute angles at C 
and E (Theorem 22) which are equal 

B 


to one another, as we can easily see 4 D 
by thinking the quadrilateral super- Fig. 11. 
imposed upon itself so that the line BE falls upon upon AC and AC 
upon BE. 

Halve AB and erect at the mid-point D the line DF perpendicular to 
AB. This line must also be perpendicular to CH, since the quadrilat- 
erals CADF and FDBE fit one another if we so place one on the other 
that the line DF remains in the same position. Hence the line CE can 
not be parallel to AB, but the parallel to AB for the point C, namely 
CG, must incline toward AB (Theorem 16) and cut from the perpendic- 
ular BE a part BG < CA. 

Since C isa random point in the line CG, it follows that CG itself 
nears AB the more the farther it is prolonged. 


22 : THEORY OF PARALLELS. 


25. Two straight lines which are parallel to a third are also parallel to 


one another. 





Wiey 42) 


We will first assume that the three lines AB, CD, EF (Fig. 12) lie in 
one plane. If two of them in order, AB and CD, are parallel to the 
outmost one, EF’, so are AB and CD parallel to one another. In order 
to prove this, let fall from any point A of the outer line AB upon the 
other outer line FE, the perpendicular AEH, which will cut the middle 
line CD in some point C (Theorem 3), at an angle DCE < }z on the 
side toward EF, the parallel to CD (Theorem 22). 

A perpendicular AG let fall upon CD from the same point, A, must 
fall within the opening of the acute angle ACG (Theorem 9); every 
other line AH from A drawn within the angle BAC must cut EF, the 
parallel to AB, somewhere in H, how small soever the angle BAH may 
be; consequently will CD in the triangle AEH cut the line AH some- 
where in K, since it is impossible that it should meet EF. If AH from 
the point A went out within the angle CAG, then must it cut the pro- 
. longation of CD between the points C and G in the triangle CAG. 
Hence follows that AB and CD are parallel (Theorems 16 and 18). 

Were both the outer lines AB and EF assumed parallel to the middle 
line CD, so would every line AK from the point A, drawn within the 
angle BAH, cut the line CD somewhere in the point K, how small soever 
the angle BAK might be. 

Upon the prolongation of AK take at random a point L and join it 


THEORY OF PARALLELS. 93 


with C by the line CL, which must cut EF somewhere in M, thus mak- 
ing a triangle MCE. 

The prolongation of the line AL within the triangle MCE can cut 
neither AC nor CM asecond time, consequently it must meet EI some- 
where in H; therefore AB and EF are mutually parallel. 


A_G H 





Bree 3; 


Now let the parallels AB and CD (Fig. 13) lie in two planes whose 
intersection line is EF. From a random point E of this latter let 
fall a perpendicular EA upon one of the two parallels, e. g., upon AB, 
then from A, the foot of the perpendicular EA, let fall a new perpen- 
dicular AC upon the other parallel CD and join the end-points E and C 
of the two perpendiculars by the line EC. The angle BAC must be 
acute (Theorem 22), consequently a perpendicular CG from C let fall 
upon AB meets it in the point G upon that side of CA on which the 
lines AB and CD are considered as parallel. 

Every line EH [in the plane FEAB], however little it diverges from 
HF, pertains with the line EC to a plane which must cut the plane of 
the two parallels AB and CD along some line CH. This latter line cuts 
AB somewhere, and in fact in the very point H which is common to all 
three planes, through which necessarily also the line EH goes; conse- 
quently EF is parallel to AB. | 

In the same way we may show the parallelism of EF and CD, 

Therefore the hypothesis that a line EF is parallel to one of two other 
parallels, AB and CD, is the same as considering EF as the intersection 
of two planes in which two parallels, AB, CD, lie. 

Consequently two lines are parallel to one another if they are parallel 
to a third line, though the three be not co-planar. 

The last theorem can be thus expressed: 

Three planes intersect in lines which are all parallel to each other of the 


parallelism of two is pre-supposed, 


24 THEORY OF PARALLELS. 


26. Triangles standing opposite to one another on the sphere are equiva- 
lent in surface. ‘ 

By opposite triangles we here understand such as are made on both 
sides of the center by the intersections of the sphere with planes; in such 
triangles, therefore, the sides and angles are in contrary order. 

In the opposite triangles ABC and A’B’C’ (Fig. 14, where one of 
them must be looked upon as represented turned about), we have the 
sides AB — A’B’, BC = B’C’, CA =C’A’, and the corresponding angles 


B! 





AY 
Fig. 14. 


at the points A, B, Care likewise equal to those in the other io at 
the points A’, B’, C’. 

Through the three points A, B, C, suppose a plane passed, and upon 
it fromthe center of the sphere a perpendicular dropped whose pro- 
longations both ways cut both opposite triangles in the points D and D/ 
of the sphere. The distances of the first D from the points ABC, in 
arcs of great circles on the sphere, must be equal (Theorem 12) as well 
to each other as also to the distances D’A’, D’B’, D’C’, on the other 
triangle (Theorem 6), consequently the isosceles triangles about the points 
D and D’ in the two spherical triangles ABC and A’B’C’ are congruent. 

In order to judge of the equivalence of any two surfaces.in general, 
I take the following theorem as fundamental: 

Two surfaces are equivalent when they arise from the mating or separating 
of equal parts. 


27. A three-sided solid angle equals the half sum of the surface angles 
less a right-angle. 

In the spherical triangle ABC (Fig. 15), where each side < z, desig. 
nate the angles by A, B, C; prolong the side AB so that a whole circle 
ABA’B’A is produced; this divides the sphere into two equal parts. 


THEORY OF PARALLELS. 25 


In that half in which is the triangle ABO, prolong now the other two 
sides through their common intersection point C until wey meet the 
circle in A’ and B’. 





C. 
Fia. 15. 


In this way the hemisphere is divided into four triangles, ABC, ACB’, 
B’/CA’, A’CB, whose size may be designated by P, X, Y, Z. It is evi. 
dent that here P+ X=—B, P+ Z=— A. 

The size of the spherical triangle Y equals that of the opposite triangle 
ABC’, having a side AB in common with the triangle P, and whose 
third angle C’ lies at the end-point of the diameter of the sphere which 
goes from C through the center D of the sphere (Theorem 26). Hence 
it follows that 

I aa Y =O, and since P + X + Y + Z=—7z, therefore we have also 

| P(A) B+ C— x). 

We may attain to the same conclusion in another way, based solely 
upon the theorem about the equivalence of surfaces given above. (Theo-* 
rem 26.) | 

In the spherical triangle ABC (Fig. 16), halve the sides AB and BQ, 
and through the mid-points D and 
E draw a great circle; upon this let 
fall from A, B, C the perpendiculars 
AF’, BH, and CG. If the perpendic- » 
ular from B falls at H between D and 
H, then will of the triangles so made 
BDH = AFD, and BHE — EGC (The. 4 
orems 6 and 15), whence follows that Fic. 16, 
the surface of the triangle ABC equals that of the quadrilateral AFGC 
(Theorem 26). 





26 THEORY OF PARALLELS. 


If the point H coincides with the middle point E of the side BC (Fig. 
B 17), only two equal right-angled triangles, ADF 
and BDE, are made, by whose interchange the 


r R equivalence of the surfaces of the triangle ABC 
and the quadrilateral AFEC is established. 

If, finally, the point H falls outside the triangle 

A ° ABC (Fig. 18), the perpendicular CG goes, in 

Fig. 17. consequence, through the triangle, and so we go 


over from the triangle ABC to the quadrilateral AFGC by adding the 


B 


A 0 
Ita. 18. 
triangle FAD — DBH, and then taking away the triangle CGE = EBH. 

Supposing in the spherical quadrilateral AF'GO a great circle passed 
through the points A and G, as also through F and OC, then will their 
arcs between AG and FC equal one another (Theorem 15), consequently 
also the triangles FAC and ACG be congruent (Theorem 15), and the 
angle FAC equal the angle ACG. ‘ 

Hence follows, that in all the preceding cases, the sum oot all three 
angles of the spherical triangle equals the sum of the two equal angles 
in the quadrilateral which are not the right angles. 

Therefore we can, for every spherical triangle, in which the sum of 
the three angles is 8, find a quadrilateral with equivalent surface, in 
which are two right angles and two equal perpéndicular sides, and 

where the two other angles are each 48. 


THEORY OF PARALLELS. ay 


Let now ABCD (Fig. 19) be the spherical quadrilateral, where the 
sides AB— DC are perpendicular to BC, and the angles A and D 
each 48. 


P 


A D 
Fig. 19. 


Prolong the sides AD and BC until they cut one another in E, and 
further beyond E, make DE = EF and let fall upon the prolongation 
of BC the perpendicular FG. Bisect the whole arc BG and join the 
mid-point H by great-circle-arcs with A and F. 

The triangles EFG and DCE are congruent (Theorem 15), so FG = 
DE AB. 

The triangles ABH and HGF are likewise congruent, since they are 
right angled and have equal perpendicular sides, consequently AH and 
AF pertain to one circle, the are AHF =z, ADEF likewise — 7, the 
angle HAD = HFE— 48S — BAH=— 48S — HFG = 48 — HFE—EFG 
—=45S—HAD—7z-+45; consequently, angle HFE=4(S—z); or what 
is the same, this equals the size of the lune AHFDA, which again is 
equal to the quadrilateral ABCD, as we easily see if we pass over from 
the one to the other by first adding the triangle EFG and then BAH 
and thereupon taking away the triangles equal to them DCE and HFG. 

Therefore $(S—z) is the size of the quadrilateral ABCD and at the 
same time also that of the spherical triangle in which the sum of the 
three angles is equal to S. 


28 THEORY OF PARALLELS. 


28. If three planes cut each other in parallel lines, then the swm of the 
three surface angles equals two right angles. 

Let AA’, BB’ CC’ (Fig. 20) be three parallels made by the inter- 
section of planes (Theorem 25). Take upon them at random three 





Fig. 20. 
points A, B, C, and suppose through these a plane passed, which con- 


sequently will cut the planes of the parallels along the straight lines 
AB, AC, and BC. Further, pass through the line AC and any point 
D on the BB’, another plane, whose intersection with the two planes of 
the parallels AA’ and BB’, CC’ and BB’ produces the two lines AD 
and DO, and whose inclination to the third plane of the parallels AA’ 
and CC’ we will designate by w. 

The angles between the three planes in which the parallels lie will 
be designated by X, Y, Z, respectively at the lines AA’, BB’, CC’; 
finally call the linear angles BDC = a, ADC = b, ADB=c. 

About A as center suppose a sphere described, upon which the inter- 
sections of the straight lines AC, AD AA’ with it determine a spherical 
triangle, with the sides p, q, and r. Call its size z Opposite the side 
q lies the angle w, opposite r lies X, and consequently opposite p lies 
the angle 7-+24—w—X, (Theorem 27). ~ 

In like manner CA, CD, CC’ cut a sphere about the center OC, and 
determine a triangle of size 8, with the sides p’, q’, r’, and the angles, w 


opposite q’, Z opposite r’, and consequently z-++-23—w—Z opposite p’. - 


Finally is determined by the intersection of a sphere about D with 
the lines DA, DB, DC, a spherical triangle, whose sides are 1, m, n, and 
the angles opposite them w+Z—2§, w+X—2a, and Y. Consequently 
its size 6 = 4 (X+Y +-Z—7)—a—B-+ W. 


Decreasing w lessens also the size of the triangles g and #, so that . 


a+ f—w can be made smaller than any given number. 


oad 


THEORY OF PARALLELS, 29 


In the triangle can likewise the sides 1 and m be lessened even to 
vanishing (Theorem 21), consequently the triangle d can be placed with 
one of its sides | or m upon a great circle of the sphere as often as you 
choose without thereby filling up the half of the sphere, hence @ van- 
ishes together with w; whence follows that necessarily we must have 

| X+Y+2Z4—=rn7 

29. In a rectilineal triangle, the perpendiculars erected at the mid-potnts 
of the sides either do not meet, or they all three cut each other in one point. 

Having pre-supposed in the triangle ABO (Fig. 21), that the two per- 
pendiculars ED and DF, which are erected upon the sides AB and BC 
at their mid points E and F, intersect in the point D, then draw within 
the angles of the triangle the lines DA, DB, DC. 

In the congruent triangles ADE and BDE (Theorem 10), we have 
AD= BD, thus follows also that. BD— CD; the 
triangle ADC is hence isosceles, consequently the 
perpendicular dropped from the vertex D upon the 
base AC falls upon G the mid point of the base. 

The proof remains unchanged also in the case 





when the intersection point D of the two perpen- 
diculars ED and FD falls in the line AC itself, or 7 £ 
falls without the triangle. Fie. 21. 

In case we therefore pre-suppose that two of those perpendiculars do 
not intersect, then also the third can not meet with them. 

30. The perpendiculars which are erected upon the sides of a rectilineal 
triangle at their mid-points, must all three be parallal to each other, so soon 
as the parallelism uf two of them %s pre-supposed, 

In the triangle ABC (Fig. 22) let the lines DH, FG, HK, be erected 
perpendicular upon the sides at their mid- 3 
points D, F, H. We will in the first place 
assume that the two perpendiculars DE and 
FG are parallel, cutting the line AB in L 
and M, and that the perpendicular HK lies 
between them. Within the angle BLE draw 
from the point L, at random, a straight line 
LG, which must cut FG somewhere in G, 
how small soever the angle of deviation GLE: may be. (Theorem 16). 





30 THEORY OF PARALLELS. 


Since in the triangle LGM the perpendicular HK can not meet with 
MG (Theorem 29), therefore it must cut LG somewhere in P, whence 
follows, that HK is parallel to DH (Theorem 16), and to MG (Theorems 
18 and 25). 

Put the side BC= 2a, AC= 2b, AB= 2c, and designate the an- 
gles opposite these sides by A, B, C, then we have in the case just 
considered 

A = IT(b)—/I(c), 

B= [I (a)—I1(¢), 

C= //(a)+ //(b), 
as one may easily show with help of the lines AA’, BB’, CC’, which 
are drawn from the points A, B, OC, parallel to the perpendicular HK 
and consequently to both the other perpendiculars DE and FG (Theo- 
rems 23 and 25). 

Let now the two perpendiculars HK and FG be parallel, then can 
the third DE not cut them (Theorem 29), hence is it either parallel to 
them, or it cuts AA’. 

The last assumption is not other than that the angle 

O> II (a)-+IT(b.) 

If we lessen this angle, so that it becomes equal to //(a)-+//(b), 
while we in that way give the line AC the new position CQ, (Fig. 23), 
and designate the size of the third side BQ by 2c’, then must the angle 
CBQ at the point B, which is increased, in accordance with what is 
proved above, be equal to 

I(a)—H(e")> H(a)—H(e), 
whence follows c’ >c (Theorem 23). 
A 


B 
Fig. 23. 


In the triangle ACQ are, however, the angles at A and Q equal, 
hence in the triangle ABQ must the angle at Q be greater than that at 
the point A, consequently is AB>BQ, (Theorem 9); that is c>c’, 

31. We call boundary line (oricycle) that curve lying in a plane for 
which all perpendiculars erected at the mid-points of chords are parallel to 


each other. 


THEORY OF PARALLELS. 31 


In conformity with this definition we can represent the generation of 
a boundary line, if we draw to a given line AB (Fig. 24) from a given 





Wie. 24, 
point A in it, making different angles CAB = //(a), chords AC = 2a; 
the end C of such a chord will le on the boundary line, whose points 


we can thus gradually determine. 

The perpendicular DE erected upon the chord AC at its mid-point D 
will be parallel to the line AB, which we will call the Awis of the bound- 
ary line. In like manner will also each perpendicular FG erected at the 
mid-point of any chord AH, be parallel to AB, consequently must this 
peculiarity also pertain to every perpendicular KL in general which is 
erected at the mid-point K of any chord CH, between whatever points 
C and H of the boundary line this may be drawn (Theorem 30). Such 
perpendiculars must therefore likewise, without distinction from AB, 
be called Awes of the boundary line. 

32. A circle with continually increasing radius merges into the boundary 
line. 

Given AB (Fig. 25) a chord of the boundary line; draw from the 
end-points A and B of the chord two axes 
AC and BF, which consequently will 
make with the chord two equal angles 
BAC — ABH —@ (Theorem 31). 

Upon one of these axes AC, take any- 





where the point E as center of a circle, 
and draw the arc AF from the initial point 
A of the axis AC to its intersection point 
F with the other axis BF. 

The radius of the circle, FE, corresponding to the point F will make 
on the one side with the chord AF an angle AFH —§, and on the 





32 THEORY OF PARALLELS. 





other side with the axis BF, the angle EFD—y,. It follows that the 
angle between the two chords BAF — g—3<§+7—a (Theorem 22); 
whence follows, a—B<4y. 

Since now however the angle y approaches the limit zero, ag well in 
consequence of a moving of the center E in the direction AC, when F 
remains unchanged, (Theorem 21), as also in consequence of an ap- 
proach of F to B on the axis BF, when the center E remains in its 
position (Theorem 22), so it follows, that with such a lessening of the 
angle y, also the angle a—, or the mutual inclination of the two chords 
AB and AF’, and hence also the distance of the point B on the bound- 
ary line from. the point F on the circle, tends to vanish. | 

Consequently one may also call the boundary-line a circle with i- 
Jinitely great radius. 

33. Let AA’ — BB’= « (Figure 26), be two lines parallel toward 
the side from A to A’, which parallels serve 8 


as axes for the two boundary arcs (arcs on i iy 
two boundary lines) AB==s, A’B’==s’, then is 

7 S86 4% A A 
where € is independent of the arcs s, s’ and of Fie. 26. 


the straight line x, the distance of the arc s’ from s. 

In order to prove this, assume that the ratio of the are s to s’ is 
equal to the ratio of the two whole numbers n and m. 

Between the two axes AA’, BB’ draw yet a third axis OO’, which 
so cuts off from the arc AB a part AC=—¢ and from the arc A’B’ on 
the same side, a part A’C’=?’. Assume the ratio of ¢ to s equal to 
that of the whole numbers p and g, so that: 


Divide now s by axes into ng equal parts, then will there be mg such 
parts on s/ and np on t. 7 

However there correspond to these equal parts on s and ¢ likewise 
equal parts on s’ and 7’, consequently we have 


y/ s! 


Hence also wherever the two arcs ¢ and t’ may be taken between the 
two axes AA’ and BB’, the ratio of ¢ to ¢’ remains always the same, as 


THEORY OF PARALLELS. 83 


long as the distance « between them remains the same. If we there- 
fore for x1, put s= es’, then we must have for every x 
Sia SO ch 

Since e€ is an unknown number only subjected to the condition e>1, 
and further the linear unit for x may be taken at will, therefore we may, 
for the simplification of reckoning, so choose it that by e is to be un- 
derstood the base of Napierian logarithms. 

We may here remark, that s’—0 for x= wo, hence not only does 
the distance between two parallels decrease (Theorem 24), but with the 
prolongation of the parallels toward the side of the parallelism this at 
last wholly vanishes. Parallel lines have therefore the character of 
asymptotes. 

34. Boundary surface (orisphere) we call that surface which arises 
from the revolution of the boundary line about one of its axes, which, 
together with all other axes of the boundary-line, will be also an axis 
of the boundary-surface. 

A chord is inclined at equal angles to such axes drawn through its end- 
points, wheresoever these two end-points may be taken on the boundary-surface. 


Let A, B, C, (Fig. 27), be three points on the boundary-surface; 





Citar 
, Via. 27. 
AA’, the axis of revolution, BB’ and CC’ two other axes, hence AB 
and AC chords to which the axes are inclined at equal angles A/AB 
—=B/BA, A’AC —C’CA (Theorem 31.) 


34 THEORY OF PARALLELS. 


Two axes BB’, CC’, drawn througn the end-points of the third chord 
BC, are likewise parallel and lie in one plane, (Theorem 25). 

A perpendicular DD/ erected at the mid-point D of the chord AB 
and in-the plane of the two parallels AA’, BB’, must be parallel to the 
three axes AA’, BB’, CC’, (Theorems 23 and 25); just such a perpen- 
dicular EK’ upon the chord AC in the plane of the parallels AA’, CC’ 
will be parallel to the three axes AA’, BB’, CC’, and the perpendicular 
DD’. Let now the angle between the plane in which the parallels A.A’ 
and BB’ lie, and the plane of the triangle ABC be designated by // (a), 
where a may be positive, negative or null. If a@ is positive, then erect 
FD —a within the triangle ABC, and in its plane, perpendicular upon 
the chord AB at its mid-point D. 

Were a a negative number, then must FD = a be drawn outside the 
triangle on the other side of the chord AB; when a—0, the point F 
coincides with D. 

In all cases arise two Agee right-angled triangles AFD and DFR, 
consequently we have FA = FB. 

Hrect now at I the line Fi’ perpendicular to the plane of the tri. 
angle ABC. 

Since the angle D/DF = //(a), and DF —a, so FF’ is parallel to 
DD’ and the line EE’, with which also it lies in one plane perpendicu- 
lar to the plane of the triangle ABC.. 

Suppose now in the plane of the parallels EK’, FF’ upon HF the per- 
pendicular EK erected, then will this be also at right angles to the plane 
of the triangle ABC, (Theorem 13), and to the line AE lying in this 
plane, (Theorem 11); and consequently must AE, which is perpendicu- 
lar to EK and EE’, be also at the same time perpendicular to FH, 
(Theorem 11). The triangles AEF and FEC are congruent, since they 
are right-angled and have the sides about the right angles equal, hence is 

Ane ECs i 

A perpendicular from the vertex F of the isosceles triangle BFC let 
fall upon the base BC, goes through its mid-point G; a plane passed 
through this perpendicular FG and the line FF’ must be perpendicular 
to the plane of the triangle ABC, and cuts the plane of the parallels 
BB’, CC’, along the line GG’, which is likewise parallel to BB’ and 
CC’, (Theorem 25); since now OG is at right angles to FG, and hence 
at the same time also to GG’, so consequently is ee angle C’OG 
= B’/BG, (Theorem 23). 


a  - 


THEORY OF PARALLELS. 35 


Hence follows, that for the boundary-surface each of the axes may 
be considered as axis of revolution. 

Principal-plane we will call each plane passed through an axis of the 
boundary surface. 

Accordingly every Principal-plane cuts the boundary-surface in the 
boundary line, while for another position of the cutting plane this in- 
tersection is a circle. 

Three principal planes which mutually cut each other, make with 
each other angles whose sum is z, (Theorem 28). 

These angles we will consider as angles in the boundary-triangle 
whose sides are arcs of the boundary-line, which are made on the bound: 
ary surface by the intersections with the three principal planes. Con- 
sequently the same interdependence of the angles and sides pertains to 
the boundary-triangles, that is proved in the ordinary geometry for the 
rectilineal triangle. 

35. In what follows, we will designate the size of a line by a letter 
‘with an accent added, e. g. x’, in order to indicate that this has a rela, 
tion to that of another line, which is represented by the same letter 
without accent x, which relation is given by the equation 

IT (2) + I(x!) = $7. 

Let now ABC (Fig. 28) be a rectilineal right-angled triangle, where 

the hypothenuse AB =e, the other sides AC=—b, BC =a, and the 





Fig. 28. 


angles opposite them are 
BAC = /[I(a), ABC = /T(§). 


36 THEORY OF PARALLELS. 


At the point A erect the line AA’ at right angles to the plane of the 
triangle ABC, and from the points B and C draw BB’ and CC’ parallel 
to AA’, 

The planes in which these three parallels lie make with each other 
the angles: //(z) at AA’, a right angle at CC’ (Theorems 11 and 13), 
consequently //(z’) at BB’ (Theorem 28). 

The intersections of the lines BA, BC, BB’ with a sphere described 
about the point B as center, determine a spherical triangle mnk, in which 
the sides are mn = [](c),. kn= I1(8), mk— [](a) and the opposite angles 
are [I(b), IM(u!), 42. 

Therefore we must, with the existence of a rectilineal triangle whose 
sides are a, b, c and the opposite angles // (z), //() 47, also admit the 
existence of a spherical triangle (Fig. 29) with the sides //(c), /7({), 
[](a) and the opposite angles //(b), //(a’), $z- 


Tc 





p) 
Fig. 29. 

Of these two triangles, however, also inversely the existence of the 
spherical triangle necessitates anew that of a rectilineal, which in con- 
sequence, also can have the sides a, a’, 8, and the oppsite angles /](b’), 
Ic), 3x. 

Hence we may pass over from a, b, ¢, 2, [, to b, a, c, 8, a, and also toa, 
hs san ht yp ae | 

Suppose through the point A (Fig. 28) with AA’ as axis, a bound- 
ary-surface passed, which cuts the two other axes BB’, CC’ , in B” and 
C”, and whose intersections with the planes the parallels form a bound- 
ary-triangle, whose sides are B’C” =p, C”A=gq, B”A=7, and the 
angles opposite them //(2), //(a’), 4x, and where consequently (Theo- 
rem 34): 

p—rsin /[](a), g=rcos [](a): 

Now break the connection of the three principal-planes along the line 
BB’, and turn them out from each other so that they with all the lines 
lying in them come to lie in one plane, where consequently the arcs p, 
q, r will unite to a single arc of a boundary-line, which goes through the | 


THEORY OF PARALLELS. yl 


point A and has AA’ for axis, in such a manner that (Fig. 30) on the 
one side will lie, the arcs g and p, the side b of the triangle, which is 





Fic. 30. 
perpendicular to AA’ at A, the axis CC’ going from the end of b par- 


allel to AA’ and through C’” the union point of p and g, the side a per- 
pendicular to CC’ at the point C, and from the end-point of a the axis 
BB’ parallel to AA’ which goes through the end-point B” of the arc p. 

On the other side of AA’ will lie, the side c perpendicular to AA’ at 
the point A, and the axis BB’ parallel to AA’, and going through the 
end-point B” of the arc r remote from the end point of b. 

The size of the line CC” depends upon b, which dependence we will 
express by CC” — f(b). 

In like manner we will have BB” — / (c). 

If we describe, taking CC’ as axis, a new boundary line from the 
point CO to its intersection D with the axis BB’ and designate the arc 
CD by ¢, then is BD = (a). 

BB’ — BD+ DB’ — BD-+CC’, consequently 
#(c)=F(a)-+ £(0). 
Moreover, we perceive, that (Theorem 32) 
| t—pet) —r sin [](a) ef), 

If the perpendicular to the plane of the triangle ABC (Fig. 28) were 
erected at B instead of at the point A, then would the lines c andr remain 
the same, the arcs g and ¢ would change to ¢ and gq, the straight lines a 


38 THEORY OF PARALLELS. 


and b into b and a, and the angle //(z) into //(f), consequently we 
would have 
q=rsin //(f) e%™, 
whence follows by substituting the value of 4g, 
cos /} (z) = sin /T(f) e%™, 
and if we change «and f into b’ and ¢, 
sin //(b) =sin //(c)e™; 
further, by multiplication with e/) 
sin /7 (b) ef) = sin [J (c) es 
Hence follows also 
sin // (a) e/) — sin /7(b) ef), 

Since now, however, the straight lines a and b are independent of 
one another, and moreover, for b—0, f(b)=—0, //(b)—47, so we have 
for every straight line a 

e—f@) —sin /] (a). 

Therefore, 

sin // (c) sin // (a) sin /7(b), 
sin []() —cos // (a) sin // (a). 
Hence we obtain besides by mutation of the letters 
sin // (a) —cos //() sin /7(b), 
cos [](b) = cos [/ (c) cos J] (2), 
cos [] (a) cos /](c) cos /7({). 

If we designate in the right-angled spherical triangle (Fig. 29) the 
sides /J(c), //(), [7 (a), with the opposite angles //(b), //(a’), by the 
letters a, b, c, A, B, then the obtained equations take on the form of 
those which we know as proved in spherical trigonometry for the right- 
angled triangle, namely, 3 

sin a=sin c sin A, 

sin b=sin ¢ sin B, 

cos A=cos asin B, 

cos B=cos b, sin A, 

COs C=Cos a, cos b; 
from which equations we can pass over to those for all spherical tri- 
angles in general. 

Hence spherical trigonometry is not dependent upon whether in a 


THEORY OF PARALLELS. 39 


rectilineal triangle the sum of the three angles is equal to two right 
angles or not. 

86. We will now consider anew the right-angled rectilineal triangle 
ABC (Fig. 31), in which the sides are a, b, c, and the opposite angles 
I(x), I1(8), ¥7- 

Prolong the hypothenuse c through 
the point B, and make BD=§; at the 
point D erect upon BD the perpendicu- 
lar DD’, which consequently will be 
parallel to BB’, the prolongation of the 
side a beyond the point B. Parallel to 
DD’ from the point A draw AA’, which 
is at the same time also parallel to CB’, 
(Theorem 25), therefore is the angle 

A/AD=II (e+) 
A’AC= J](b), consequently 
IT(b)= II (a) +I (c+) 


B If from B we lay off 8 on the hypoth- 
enuse c, then at the end point D, (Fig. 
32), within the triangle erect upon AB 
the perpendicular DD’, and from the 
point A parallel to DD’ draw AA’, so 
will BC with its prolongation CC’ be the 
third parallel; then is, angle CAA’=// 
(b), DAA’= /T (c—8), consequently //(c— 
6)=/T(a)+/1(b). The last equation is 
then also still valid, when c=, or c<f. 

. D If c= (Fig. 33), then the perpendicu- 
Fig. 32. ular AA’ erected upon AB at the point A 








40 THEORY OF PARALLELS. 


is parallel to the side BC=a, with its prolongation, CC’, consequently 





Fiq. 33. 


we have //(«)+//(b)= 37, whilst also //(c—f)=4z, (Theorem 23). 

If c<{3, then the end of f falls beyond the point A at D (Fig. 34) 
upon the prolongation of the hypothenuse AB. Here the perpendicu- 
lar DD’ erected upon AD, and the line AA’ parallel to it from A, will 
likewise be parallel to the side BC=a, 
with its prolongation CC’. 

Here we have the angle DAA’ = // 
(§—c), consequently 
[M(a)-+ IIb) =z—IN(p—e)= M(e—P) 
(Theorem 23). 

The combination of the two equations 
found gives, ; 

211(b)=M(c—B) + Ile), 
21(a)=1(c—f)— H+ 8), 
whence follows 
cos [I(b) cos [ 4/e—p)+4 Mce+8)] 
cos [I(x) cos [ 4/(e—f)—4 I-A) 
Substituting here the value, (Theo- 
rem 35) | 
cos IT (b) 
Fig. 34. == COB TC), 
cos [7 (z) 
we have [tan 4/7 (c) ?—=tan 4/7 (c—f) tan 4// (c+). 
Since here § is an arbitrary number, as the angle //(/%) at the one 








THEORY OF PARALLELS. 4] 


side of c may be chosen at will between the limits 0 and 47, conse- 
quently § between the limits 0 and o, so we may (leduce by taking 
consecutively G—c, 2c, 3c, &c., that for every positive number n, [tant 
II(c)|»=tan4 /](nc). 

If we were nas the ratio of two lines x and c, and assume that 
cots [](c)= 
then we ve ue every line x in general, Preiion it be positive or nega- 
tive, tan} /](x)—e—* 
where e may be any arbitrary number, which is greater than unity, 
since [](x)=0 for x= 

Since the unit by which the-lines are measured is arbitrary, so we 
may also understand by e the base of the Napierian Logarithms. 
_ 87. Of the equations found above in Theorem 35 it is sufficient to 
know the two following, 

sin //(c)—.sin /] (a) sin //(b) 
sin // (z)—sin /[/(b) cos //({), 

applying the latter to both the sides a and b about the right angle, in 
order from the combination to deduce the remaining two of Theorem 
35, without ambiguity of the algebraic sign, since here all angles are 
acute. 

In a similar manner we attain the two equations 

(1.) tan //(c)=sin //(z) tan //(a), 
(2.) cos [/(a)—cos [/(c) cos /I (8). 


We will now consider a rectilineal triangle whose sides are a, b, c, 
(Fig. 35) and the opposite angles A, B, C. 

If A and B are acute angles, then the 
perpendicular p from the vertex of the 
angle C falls within the triangle and cuts 
the side c into two parts, x on the side of 
| : the angle A and c—x on the side of the 
Fie. 35. angle B. Thus arise two right-angled 





triangles, for which we obtain, by application of equation (1), 
tan //(a)=sin B tan //(p), 
tan //(b)=sin A tan //(p), 


49 THEORY OF PARALLELS. 


which equations remain unchanged also when one of the angles, e. q. B, 
is a right angle (Fig. 36) or and obtuse angle (Fig. 37). 


C C 

b > 

A — B A C B r 
Fia. 36. Fie. 37. 


Therefore we have universally for every triangle 
(3.) sin A tan //(a)=sin B tan //(b). 

For a triangle with acute angles A, B, (Fig. 35) we have also (Equa- 

tion 2), 

cos /](x)=cos A cos //(b), 

cos //(c—x)=cos B cos //(a) | 
which equations also relate to triangles, in which one of the angles A 
or B is a right angle or an obtuse angle. 

As example, for B=4z (Fig. 36) we must take x=c, the first equa- 
tion then goes over into that which we have found above as Equation 2, 
the other, however, is self-sufficing. 

For B>4z (Fig. 37) the first equation remains unchanged, instead 
of the second, however, we must write correspondingly 

cos [](x—c)=cos (z—B) cos //(a); 
but we have cos //(x—c)=—cos /[](c—x) 
(Theorem 23), and also cos (t—B)=—cos B. 

If A is a right or an obtuse angle, then must c—x and x be put for 
x and c—x, in order to carry back this case upon the preceding. 

In order to eliminate x from both equations, we notice that (Theo- 
rem 36) 
1—[tan} {,\c—x) ]? 
1-+[tan 4//(c—x) |? 
1—e2x— 2¢ 
=P sseouas 

1—[tan 4//(c)]?[cot £//(x)]? 

cos [1(c)—cos/1(x) 
~~ 1—cos II(c)cos [1(x) 





cos /](c—x)= 


THEORY OF PARALLELS. 43 


If we substitute here the expression for cos //(x), cos //(c—x), we ob- 
tain 
cos /I(a) cos B+-cos/I(b) cos A 


cos [](¢)— 1+¢0s //(a) cos //(b) cosA cosB 





whence follows 


cos [I(c)—cos cos /(b) 


B= 3 [Tc 
cos [](a) cos 1—cosA cos/1(b) cos [/(c) 





and finally 
[sin /7 (c) ]2 =[1—cos Boos /](c) cos /] (a) ][1—cos A cos /] (b) cos JT (c) ] 
In the same way we must also have 
(4. 

[sin // (a) ]? =[1—cos C cos /] (2) cos /] (b) ][1—cos B cos /] (c) cos /] (a) | 
[sin IT(b) ]? =[1—cos A cos /] (b) cos /7(c) ][1—cos C cos /7 (a) cos /7(b) | 
From these three equations we find 
[sin /7(b)]? [sin (c)]? 
[sin 11 (a) 


Hence follows without ambiguity of sign, 





=[1—cosA cos //(b) cos II(c)] 2 


: : sin // (b)sin // (c 
(5.) cos A cos IT(b) cos /1(c) $a ia 
If we substitute here the value of sin //(c) corresponding to equa- 
tion (3.) 
: sin A. 
he sin C 





tan // (a) cos [7 (c) 
then we obtain 
cos // (a) sin C 
sin A sin //(b)+-cos A sin C cos// (a) cos /7(b); 
but by substituting this expression for cos //(c) in equation (4), 


cos IT (c) —= 





(6.) cot A sinC sin /] Micee cae 
By elimination of sin //(b) with help of the equation (3) comes 
cos [T (a) 7; cosA 
oe TE) cos C=1 — ——sysinC sin II (a). 
In the meantiine the equation (6) gives by changing the letters, 
cos IT (a) 


cos II(b) —cot B sinC sin // (a)-+-cosC. 


44 THEORY OF PARALLELS. 


From the last two equations follows, 
sin B sinC 
sin [/ (a) 


All four equations for the interdependence of the sides a, b, c, and 


(7.) cos A-+-cos B cosC— 


the opposite angles A, B, C, in the rectilineal triangle will therefore be, 


[ Equations (3), (5), (6), (7).] 
/sin A tan //(a) = sin B tan // (b), 








sin /] (b) sin //(c) 
cos A cos //(b) cos /[(c) + — sin [7 (a) =a 
(8.) eee A sin C sin /] (b) + cos C =e 
sin BsinC 
cos A + cos Bcos C = mT 


If the sides a, b, c, of the triangle are very small, we may content our- 
selves with the approximate determinations. (Theorem 36.) | 


cot /] (a) = a, 
sin // (a) = 1 — $a? 
cos /] (a) = a, 


and in like manner also for the other sides b and c. 
The equations 8 pass over for such triangles into the following: 
bsin A = asin B, 
a2 =b2 + c? — 2becosA, 
asin (A -++ C) = bsin A, 
cos A. + cos(B + C) = 0. ) 

Of these equations the first two are assumed in the ordinary geom- 

etry; the last two lead, with the help of the first, to the conclusion 
A+B+C=rnz. 

Therefore the imaginary geometry passes over into the ordinary, when 
we suppose that the sides of a rectilineal triangle are very small. 

I have, in the scientific bulletins of the University of Kasan, pub. 
lished certain researches in regard to the measurement of curved lines, 
of plane figures, of the surfaces and the volumes of solids, as well as in 
relation to the application of imaginary geometry to analysis. 

The equations (8) attain for themselves already a sufficient foundation 
for considering the assumption of imaginary geometry as possible. 
Hence there is no means, other than astronomical observations, to use 


THEORY OF PARALLELS. 45 


_ for judging of the exactitude which pertains to the calculations of the 
ordinary geometry. : 

This exactitude is very far-reaching, as | have shown in one of my 
investigations, so that, for example, in triangles whose sides are attain- 
able for our measurement, the sum of the three angles is not indeed dif- 
ferent from two right angles by the hundreath part of a second. 

In addition, it is worthy of notice that the four equations (8) of 
plane geometry pass over into the equations¢for spherical triangles, if 
we put a,/— 1, b ,/— 1, c,/— 1, instead of the sides a, b, c; with this 
change, however, we must also put 





. 1 
sin // (a) Babee (5) 
cos [7 (a) = (4/—1) tana, 
gtd eres ae 1), . 


and similarly also for the sides b and c. 
In this manner we pass over from equations (8) to the following: 
sin A sin b = sin Bsina, 
cosa = cos b cose + sin b sinc cos A, 
cot A sin C + cosC cos b = sin b cota, 
cos A = cosa sin B sin C — cos B cosC. 





TRANSLATOR’S APPENDIX. 


ELLIPTIC GEOMETRY. 


Gauss himself never published aught upon this fascinating subject, 
Geometry Non-Euclidean; but when the most extraordinary pupil of 
his long teaching life came to read his inaugural dissertation before the 
Philosophical Faculty of the University of Goettingen, from the three 
themes submitted it was the choice of Gauss which fixed upon the one 
“Ueber die Hypothesen welche der Geometrie zu Grunde liegen.” 

Gauss was then recognized as the most powerful mathematician in the 
world. I wonder if he saw that here his pupil was already beyond him, 
when in his sixth sentence Riemann says, ‘therefore space is only a 
special case of a three-fold extensive magnitude,” and continues: 
“From this, however, it follows of necessity, that the propositions of 
geometry can not be deduced from general magnitude ideas, but that 
those peculiarities through which space distinguishes itself from other 
thinkable threefold extended magnitudes can only be gotten from ex- 
perience. Hence arises the problem, to find the simplest facts from 
which the metrical relations of space are determinable—a problem 
which from the nature of the'thing is not fully determinate; for there 
may be obtained several systems of simple facts which suffice to deter- 
mine the metrics of space; that of Huclid as weightiest is for the pres- 
ent aim made fundamental. These facts are, as all facts, not necessary, 
but only of empirical certainty; they are hypotheses. Therefore one 
can investigate their probability, which, within the limits of observation, 
of course is very great, and after this judge of the allowability of their 
extension beyond the bounds of observation, as well on the side of the 
immeasurably great as on the side of the immeasurably small.” 

Riemann extends the idea of curvature to spaces of three and more 
dimensions. The curvature of the sphere is constant and positive, and 
on it figures can freely move without deformation. The curvature of 
the plane is constant and zero, and on it figures slide without stretching. 
The curvature of the two-dimentional space of Lobatschewsky and 

[47] 


48 THEORY OF PARALLELS, 


Bolyai completes the group, being constant and negative, and in it fig- 
ures can move without stretching or squeezing. As thus corresponding 
to the sphere it is called the pseudo-sphere. 

In the space in which we live, we suppose we can move without de- 
formation. It would then, according to Riemann, be a special case of 
a space of constant curvature We presume its curvature null. At 
once the supposed fact that our space does not interfere to squeeze us 
or stretch us when we move, is envisaged as a peculiar property of our 
space. But is it not absurd to speak of space as interfering with any- 
thing? If you think so, take a knife and a raw potato, and try to cut 
it into a seven-edged solid. 

Father on in this astonishing discourse comes the epoch-making idea, 
that though space be unbounded, it is not therefore infinitely great. 
Riemann says: ‘‘In the extension of space-constructions to the im- 
measurably great, the unbounded is to be distinguished from the in- 
finite; the first pertains to the relations of extension, the latter to the 
size-relations. 

“That our space is an unbounded three-fold extensive manifoldness, is 
an hypothesis, which is applied in each apprehension of the outer world, 
according to which, in each moment, the domain of actual perception is 
filled out, and the possible places of a sought object constructed, and 
which in these applications is continually confirmed. The unbounded- 
ness of space possesses therefore a greater empirical certainty than any 
outer experience. From this however the Infinity in no way follows. 
Rather would space, if one presumes bodies independent of place, that 
is ascribes to it a constant curvature, necessarily be finite so soon as this 
curvature had ever so small a positive value. One would, by extend- 
ing the beginnings of the geodesics lying in a surface-element, obtain 
an unbounded surface with constant positive curvature, therefore a sur- 
face which in a homaloidal three-fold extensive manifoldness would 
take the form of a sphere, and so is finite.” 

Here we have for the first time in human thought the marvelous per- 
ception that universal space may yet be only finite. ; 

Assume that a straight line is uniquely determined by two points, but 
take the contradictory of the axiom that a straight line is of infinite 
size; then the straight line returns into itself, but two having inter- 
sected get back to that intersection point without ever again meeting. 


TRANSLATOR’S APPENDIX. 49 


Two intersecting complete straight lines enclose a plane figure, a digon. 
Two digons are congruent if their angles are equal. All complete 
straight lines are of the same length, 7. In a given plane all the per- 
pendiculars to a given straight line intersect in a single point, whose 
distance from the straight line is 47. Inversely, the locus of all the 
points at a distance 4/ on straight lines passing through a given point 
and lying in a given plane, is a straight line perpendicular to all the 
radiating lines. 

The total volume of the universe is /3/z. 

The sum of the angles of a plane triangle is greater than a straight 
angle by an excess proportional to its area. 

The greater the area of the triangle, the greater the excess or differ- 
ence of the angle sum from z. 

Says the Royal Astronomer for Ireland: 

“It is necessary to measure large triangles, and the largest triangles 
to which we have access are, of course, the triangles which the astrono- 
mers have found means of measuring. The largest available triangles 
are those which have the diameter of the earth’s orbit as a base and a 
_ fixed star at the vertex. It is a very curious circumstance that the in- 
vestigations of annual parallax of the stars are precisely the investiga- 
tions which would be necessary to test whether ono of these mighty tri- 
angles had the sum Of its three angles equal to two rightangles. * * * 

“ Astronomers have sometimes been puzzled by obtaining a negative 
parallax as the result of their labors. No doubt this has generally or 
indeed always arisen from the errors which are inevitable in inquiries of 
this nature, but if space were really curved then a negative parallax 
might result from observations which possessed mathematical perfec- 
tion. * * * It must remain an open question whether if we had 
large cnough triangles the sum of the three angles would still be two 
right angles.” 

Says Prof. Newcomb: ‘There is nothing within our experience 
which will justify a denial of the possibility that the space in which we 
find ourselves may be curved in the manner here supposed. * * * 

“The subjective impossibility of conceiving of the relation of the 
most distant points in such a space does not render its existence in- 
credible. In fact our difficulty is not unlike that which must have been 
felt by the first man to whom the idea of the sphericity of the earth 


5O THEORY OF PARALLELS. 


was suggested in conceiving how by traveling in a constant direction 
he could return to the point from which he started without during his 
journey feeling any sensible change in the direction of gravity.” 

In accordance with Professor Cayley’s Sixth Memoir upon Quantics: 

“The distance between two points is equal io c times the logarithm of the 
cross ratio in which the line joining the two points"is divided by the funda- 
mental quadric.”’ 

This projective expression for distance, and Laguerre’s for an angle 
were in 1871 generalized by Felix Klein in his article Ueber die soge- 
nannte Nicht-EKuklidische Geometric, and in 1872 (Math. Ann., Vol. 6) 
he showed the equivalence of projective metrics with non-Euclidean 
geometry, space being of constant negative or positive curvature ac- 
cording as the fundamental surface is real and not rectilineal or is im- 
aginary. | 

We have avoided mentioning space of four or more dimensions, 
wishing to preserve throughout the synthetic standpoint. 

For a bibliography of hyper-space and non-Euclidean geometry see 
articles by George Bruce Halsted in the American Journal of Mathe- 
matics, Vol. L, pp. 261-276, 384, 385; Vol. IL., pp. 65-70. 

We notice that Clark University and Cornell University are giving 
regular courses in non-Euclidian geometry by their most eminent Pro- 
fessors, and we presume, without looking, that the same is true of Har- 
vard and the Johns Hopkins University, with Prof. Newcomb an origi- 
nal authority on this far-reaching subject. 





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